Nearly Sasakian geometry and \(SU(2)\) -structures

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• 作者：Beniamino Cappelletti-Montano ; Giulia Dileo
• 刊名：Annali di Matematica Pura ed Applicata
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：195
• 期：3
• 页码：897-922
• 全文大小：588 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1618-1891
• 卷排序：195

文摘

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian and 5-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the 5-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo \(SU(2)\)-structures. By deforming such an \(SU(2)\)-structure, one obtains in fact a Sasaki–Einstein structure. Further we prove that both nearly Sasakian and Sasaki–Einstein 5-manifolds are endowed with supplementary nearly cosymplectic structures. We show that there is a one-to-one correspondence between nearly cosymplectic structures and a special class of hypo \(SU(2)\)-structures which is again strictly related to Sasaki–Einstein structures. Furthermore, we study the orientable hypersurfaces of a nearly Kähler 6-manifold, and in the last part of the paper, we define canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly Kähler geometry. In dimension 5, we determine a connection which parallelizes all the nearly Sasakian \(SU(2)\)-structure as well as the torsion tensor field. An analogous result holds also for Sasaki–Einstein structures.KeywordsNearly SasakianSasaki–Einstein\(SU(2)\)-structureNearly cosymplecticContact manifold Nearly Kähler